On the discrepancy estimate of normal numbers

Transcription

1 ACTA ARITHMETICA LXXXVIII.2 (1999 On the discrepancy estimate of normal numbers 1. Introduction by M. B. Levin (Tel-Aviv Dedicated to Professor N. M. Korobov on the occasion of his 80th birthday 1.1. A number α (0, 1 is said to be normal to base q if in the q-ary expansion of α, α.d 1 d 2... (d i 0, 1,..., q 1}, i 1, 2,..., each fixed finite block of digits of length k appears with an asymptotic frequency of q k along the sequence (d i i 1. Normal numbers were introduced by Borel ( Let (x n n 1 be an arbitrary sequence of real numbers. The quantity (1 D(N D(N, (x n n 1 sup #0 n < N x n } < γ}/n γ γ (0,1] is called the discrepancy of (x n N n1, where x} x [x] is the fractional part of x. The sequence x n } n 1 is said to be uniformly distributed (u.d. in [0, 1 if D(N It is known that a number α is normal to base q if only if the sequence αq n } n 0 is u.d. (Wall, Borel proved that almost every number (in the sense of Lebesgue measure is normal to base q. In [G], Gal Gal proved that D(N, αq n } n 0 O((N 1 log log N 1/2 for a.e. α In [K1] Korobov posed the problem of finding a function ψ with maximum decay, such that α : D(N, αq n } n 0 ψ(n, N 1, 2, Mathematics Subject Classification: 11K16, 11K38. Work supported in part by the Israel Science Foundation Grant No [99]

2 100 M. B. Levin He showed that ψ(n O(N 1/2 (see [K1]. The lower bound of the discrepancy for the Champernowne Davenport Erdős normal numbers was found by Schiffer [S]: D(N, αq n } n 1 K/ log N with K > 0, N 2, 3,... For a bibliography on Korobov s problem see [Po, L1] In [L2] we proposed using small discrepancy sequences (van der Corput type sequences nα} n 0 to construct normal numbers, announced that ψ(n O(N 1 log 2 N. This result is proved below. The estimate of ψ(n was previously known to be O(N 2/3 log 4/3 N (Korobov [K2] for q prime, Levin [L1] for arbitrary integer q. We note that the estimate obtained cannot be improved essentially, since according to W. Schmidt, 1972 (see [N, p. 24], for any sequence of reals, lim ND(N/ log N > 0. N 1.4. Let x [a 0 (x; a 1 (x, a 2 (x,...] be the continued fraction expansion of x, with partial quotients a i (x. For an integer b Q > 1 let a i (b/q denote the sum of all partial quotients of b/q. Following [P] we prove (see Lemma 3 that there exists an integer sequence b m a constant K > 0 with m (2 ai (b m /q r } Km 3, m 1, 2,... r1 Theorem 1. Let (3 α m 1 1 q n m 0 k<q m where b m satisfy (2, (4 n 1 0 n k 1 r<k Then the number α is normal to base q, bm k q m } 1 q mk rq r, k 2, 3,... D(N, αq n } n 0 O(N 1 log 3 N Let (p i,j i,j 1 be Pascal s triangle: p i,1 p 1,i 1, i 1, 2,..., p i,j p i,j 1 + p i 1,j, i, j 2, 3,..., (p i,j i,j 1 be Pascal s triangle mod 2: (5 p i,j p i,j mod 2, i, j 1, 2,...

3 Discrepancy estimate of normal numbers 101 Every integer n 0 has a unique digit expansion in base q, (6 n j 1 e j (nq j 1 with e j (n 0,..., q 1}, j 1, 2,..., e j (n 0 for all sufficiently large j. Theorem 2. Let (7 α m 1 where 1 q n m 1 q n2m 0 n<q 2m 2 m i1 d i (n q i (8 d i (n j 1 p i,j e j (n mod q, d i (n, i 1,..., 2 m, n [0, q 2m, (9 n 1 0 n m 1 r<m Then the number α is normal to base q 2 r q 2r, m 2, 3,... D(N, αq n } n 0 O(N 1 log 2 N. Remark 1. We use here the sequence of 2 m 2 m matrices of Pascal s triangle mod 2. A similar result is valid for the sequence of m m matrices of Pascal s triangle (or m m matrices of Pascal s triangle mod p but with D(N, αq n } n 0 O(N 1 log 3 N, where α is denoted by a concatenation of blocks ω m : where α.ω 1... ω m..., ω m (d 1 (1... d m (1... d 1 (q m... d m (q m, m 1, 2,..., d i (n j 1 p i,j e j (n mod q. Remark 2. Let (σ i i 1 be any sequence of substitutions of the set 0, 1,..., q 1}. The proof of Theorem 2 does not change if in (8 we use the functions σ i (e i (n instead of the functions e i (n (see [B], [N, p. 25]. 2. Proof of the theorems. Let m 1, b, i be integers, 0 i < m, (b, q 1, α m α m (b } bk 1 (10 q m q mk, (11 0 k<q m α mni [q 2m i α m q i+mn }]/q 2m i.

4 102 M. B. Levin It is easy to see that bn/q m }q i } bn/q m i }, } } } } bn b(n b(n α m q i+mn } q m q i + q m q m i + q m q 2m i +... } } } bn b(n b(n q m i + q m q m i + q m q 2m i +... Therefore (12 α mni } bn q m i + Let N [1, mq m ] be an integer, γ (0, 1], b(n + 1 q m } 1 q m i. (13 A(γ, N, (x n #0 n < N xn } < γ} for γ > 0, 0 for γ 0, (14 A(γ, Q, P, (x n #Q n < Q + P x n } < γ}. Hence from (10 we obtain (15 A(γ, N, α m q n } n 0 A(γ, m[n/m], α m q n } n 0 + A(γ, m[n/m], N m[n/m], α m q n } n 0 m 1 i0 A(γ, [N/m], α m q i+mn } n 0 + θm with θ [0, 1]. Let c [q m γ], N 1 [1, q m ] 0 i < m. From (11 (13 we deduce (16 ( c 1 A q m, N 1, (α mni n 0 A(γ, N 1, α m q mn+i } n 0 ( c + 1 A, N 1, (α mni n 0 Lemma 1. Let N [1, mq m ] be an integer, γ (0, 1], (b, q 1. Then (17 A(γ, N, α m q n } n 0 γn + ε 1 (4m + 3 q m. m max ND(N, bn/qi } n 0, 1 N q i i1 (18 A(γ, mq m, α m q n } n 0 γmq m + 3ε 2 m with ε j < 1, j 1, 2.

5 Discrepancy estimate of normal numbers 103 P r o o f. Let 0 i < m, d, d 1, d 2 be integers, d d 1 q i + d 2, d 1 [0, q m i, d 2 [0, q i. By (12 (13 we get ( d A q m, N 1, (α mni n 0 } } bn b(n # 0 n < N 1 q m i + q m q m i < d 1 q m i + 1 } d2 qm i q i. Consequently, (19 A(d/q m, N 1, (α mni n 0 T 1 (N 1 + T 2 (N 1, where } T 1 (N # 0 n < N bn (20 q m i } T 2 (N # 0 n < N bn (21 q m i < d } 1 q m i, d 1 qm i } b(n + 1 q m < d } 2 q i. Let N 1 N 2 q m i + N 3 with N 3 [0, q m i N 2 [0, q i. It is easy to see that We see from (20 (1 that T 1 (N 1 T 1 (q m i N 2 + T 1 (N 3. (22 T 1 (N 2 q m i N 2 d 1, T 1 (N 3 d ( } 1 bn q m i N 3 + εn 3 D N 3, q m i n 0 with ε 1. This yields (23 T 1 (N 1 d ( } 1 bn q m i N 1 + ε max ND N, 1 N<q m i q m i with ε 1. n 0 Now we compute T 2 (N. Let d 0 be an integer, d 0 d 1 b 1 mod q m i with d 0 [0, q m i, Y 0 n < N 1 bn/q m i } d 1 /q m i }. Clearly if bn/q m i } d 1 /q m i, then bn d 1 mod q m i, n d 0 mod q m i, [ ] (24 Y d 0 + rq m i N1 d r < N 4 } with N Combining (21 (1 we obtain q m i

6 104 M. B. Levin } T 2 (N 1 # n Y b(n + 1 q m < d } 2 (25 q i b(d0 + 1 # 0 r < N 4 q m + br } q i < d } 2 q ( } i d 2 bn N 4 q i + ε 2N 4 D N 4, q i + θ n 0 with θ b(d 0 + 1/q m, ε 2 1. It follows from (1 that for every real θ, (26 D(N, x n + θ} n 0 2D(N, x n } n 0. By (24 (25, this yields [ N1 + q m i d 0 1 (27 T 2 (N 1 q m i ] d2 q i + 2ε 2 max ND(N, 1 N q bn/qi } n 0 i d 2 N 1 q m + ε 3 + 2ε 2 max ND(N, bn/q i } n 0 1 N q i with ε j 1, j 2, 3. If N 1 q m, then N 4 q i, N 4 D(N 4, bn/q i } n 0 1. Hence from (25 (26 we obtain (28 T 2 (q m d 2 + 2ε 4 with ε 4 1. Substituting (23 (27 into (19, we obtain A(d/q m, N 1, (α mni n 0 N 1 d/q m + ε 5 (1 + max 1 N<q m i ND(N, bn/qm i } n max 1 N q i ND(N, bn/qi } n 0 with ε 5 1. Using (16 (15 we get A(γ, N 1, α m q mn+i } n 0 A(γ, N, α m q n } n 0 θm + γn 1 + ε 6 (2 + max 1 N<q m i ND(N, bn/q m i } n max 1 N q i ND(N, bn/q i } n 0 with ε 6 1, m γ[n/m] i1 + ε 7 (2m + 3 m max ND(N, bn/qi } n 0 1 N q i i1 γn + ε 8 (4m + 3 m max ND(N, bn/q i } n 0, 1 N q i i1

7 Discrepancy estimate of normal numbers 105 where ε j 1, j 7, 8. Assertion (17 is proved. Assertion (18 follows analogously from (22 (28. Lemma 2. Let j 1, 1 N q j, (b, q 1, a i (x be partial quotients of x}. Then ND(N, bn/q j } n 0 a i (b/q j. For the proof of this well-known theorem, see for example [N, p. 26]. Lemma 3. There exists a constant K > 0 integers c m [0, q m such that m ai (c m /q r } Km 3, m 1, 2,... r1 P r o o f. According to [P, p. 2144] there exist constants K q such that ai (c/q r K q q r r 2, r 1, 2,... Therefore (29 1 c q r, (c,q1 1 c q m, (c,q1 r1 m r1 q m r m ai (c/q r } 1 c q r, (c,q1 ai (c/q r m q m K q r 2 K q q m m 3. Let φ(q m #1 c q m (c, q 1} K K q q/φ(q. It is known that φ(q m q m 1 φ(q. Now the assertion of Lemma 3 follows from (29. Corollary. Let 1 N mq m. Then r1 (30 A(γ, N, α m (b m q n } n 0 γn + O(m 3. The statement follows from (1, (2, (10, Lemmas 1 3. Applying (3 (10 we get αq n m+n } α m (b m q n } + θq n mqm with 0 < θ < 1 0 n < mq m. Hence from (13 we have, for N [1, mq m ], A(γ 1/q m, N m, α m (b m q n } n 0 A(γ, N, αq n m+n } n 0 By using (30 (14, we obtain A(γ, N, α m (b m q n } n 0. (31 A(γ, n m, N, αq n } n 0 γn + O(m 3 with 1 N mq m. Similarly, from (18 we deduce that (32 A(γ, n m, mq m, αq n } n 0 γmq m + O(m.

8 106 M. B. Levin End of the proof of Theorem 1. For every N 1 there exists an integer k such that N [n k, n k+1. By (4 this yields (33 N n k + R with 0 R < kq k, N > (k 1q k 1, k 2 log q N. Applying (4, (14 (31 (33 we obtain k 1 A(γ, N, αq n } n 0 A(γ, n r, rq r, αq n } n 0 + A(γ, n k, R, αq n } n 0 r1 k 1 (γrq r + O(r + γr + O(k 3 r1 Thus, by (1, the theorem is proved. γn + O(k 3 γn + O(log 3 N. Proof of Theorem 2. In [So] Sobol proposed the use of Pascal s triangle mod 2 to construct small discrepancy sequences (see also [F], [N]. Here we use Pascal s triangle mod 2 to construct normal numbers. Let P n be a sequence of a 2 n 2 n matrices such that P 1 ( ,..., P n+1 ( Pn P n P n 0,... It is easy to prove by induction that P n is the 2 n 2 n upper left-h corner of Pascal s triangle (5, P n is a triangular-type matrix. The following lemma is proved in [BH] for Pascal s triangle, it is clearly valid also for Pascal s triangle mod 2. Lemma 4. The determinant of any n n array taken with its first row along a row of ones, or with its first column along a column of ones in Pascal s triangle, written in rectangular form, is one. From (7 we have (34 αq n m+2 m n+k }.d k+1 (nd k+2 (n... d 2 m(nd 1 (n Let 1 k, i 2 m (35 α ki (n [αq n m+2 m n+k }q i ]/q i. It is easy to see that (36 α ki (n.dk+1 (n... d k+i (n if k + i 2 m,.d k+1 (n... d 2 m(nd 1 (n d k+i 2 m(n + 1 otherwise. Lemma 5. Let m, k, i, B, f be integers, 1 i, k 2 m, B [0, q 2m i, f [0, q i. Then A(f/q i, Bq i, q i, (α ki (n n 0 f + 2ε with ε < 1.

9 Discrepancy estimate of normal numbers 107 P r o o f. Case 1. Let k +i 2 m, c j 0, 1,..., q 1} (j 1,..., i. We examine the system of equations (37 d k+j (n c j, j 1,..., i, n [Bq i, (B + 1q i. According to (8 this system is equivalent to the system of i congruences 1 ν 2 m p k+j,ν e ν (n + Bq i c j mod q, j 1,..., i, n [0, q i. Applying (6 we see that e ν (n + Bq i e ν (n + e ν (Bq i, ν 1, 2,..., (38 p k+j,ν e ν (n 1 ν i c j with n [0, q i. It follows from Lemma 4 that (39 det(p k+j,ν 1 j,ν i 1. i<ν 2 m p k+j,ν e ν (Bq i mod q, j 1,..., i, For any c 1,..., c i the system (38 has a unique solution (e 1 (n,..., e i (n, consequently there exists a unique n 0 [Bq i, (B + 1q i satisfying (37. From (36 (37 we see that the set α ki (n n [Bq i, (B + 1q i } coincides with j/q i j [0, q i }. Hence from (14 we have (40 A(f/q i, Bq i, q i, (α ki (n n 0 f. Case 2. Let k + i > 2 m, l 1 2 m k. As in (37 (38, the system of equations (41 (42 d k+j (n c j, j 1,..., l 1, d j (n + 1 c j+l1, j 1,..., i l 1, with n [Bq i, (B + 1q i, is equivalent to the systems of congruences (43 p k+j,ν e ν (n (44 1 ν i 1 ν i c j p j,ν e ν (n + 1 c j+l1 i<ν 2 m p k+j,ν e ν (Bq i mod q, j 1,..., l 1, i<ν 2 m +1 p j,ν e ν ((B + [(n + 1/q i ]q i mod q, where j 1,..., i l 1 n [0, q i. Let n n 1 + n 2 q l 1 with n 1 [0, q l 1 n 2 [0, q i l 1. It is evident that e ν (n e ν (n 1 for ν 1,..., l 1. The matrix P m is triangular. Hence p k+j,ν 0 with ν > 2 m k j l 1 j. The system (43 is equivalent to the following system of congruences:

10 108 M. B. Levin (45 1 ν l 1 p k+j,ν e ν (n 1 c j i<ν 2 m p k+j,ν e ν (Bq i mod q, j 1,..., l 1, where n 1 [0, q l 1 n 2 [0, q i l 1. Applying (39 with i l 1 shows that this system has a unique solution with (e 1 (n 1,..., e l1 (n 1. Consequently, there exists a unique solution n 1 n 1 [0, q l 1 satisfying (45. By (41 (43 we obtain (46 (d k+1 (n 1 +n 2 q l 1 +Bq i,..., d k+l1 (n 1 +n 2 q l 1 +Bq i 0 n 1 < q l 1 } (c 1,..., c l1 c j, j 1,..., l 1 }. Now we examine the system (44 with n 1 n 1 the solution of (45. Case 2.1. Let n 1 q l 1 2. Bearing in mind that e ν (n + 1 e ν (n q l 1 n 2 e ν (n e ν (q l 1 n 2, we deduce from (44 that p j,ν e ν (q l 1 n 2 l 1 <ν i c j+l1 1 ν l 1 p j,ν e ν (n i<ν 2 m p j,ν e ν (Bq i mod q with j 1,..., i l 1 0 n 2 < q i l 1. Applying Lemma 4 we obtain a unique solution for this system with (e l1 +1(q l 1 n 2,..., e i (q l 1 n 2. By (42 (44 we get (47 (d 1 (n 1 + n 2 q l 1 + Bq i + 1,..., d i l1 (n 1 + n 2 q l 1 + Bq i n 2 < q i l 1 } (c l1 +1,..., c i c l1 +j, j 1,..., i l 1 }. Let (48 F d k+1 (n... d 2 m(nd 1 (n d k+i 2 m(n n 1 < q l 1 1, 0 n 2 < q i l 1, n n 1 + n 2 q l 1 + Bq i }, (49 g ν d k+ν (q l Bq i, ν 1,..., l 1. From (46 (47 we have (50 F (c 1,..., c i c j, j 1,..., i, (c 1,..., c l1 (g 1,..., g l1 } (51 #F q i q i l 1. Case 2.2. Let n 1 q l 1 1, n 2 [0, q i l 1 2] n n 1 + n 2 q l 1. Then e ν (n for 1 ν l 1 e ν (n + 1 e ν ((n 2 + 1q l 1 for l 1 < ν i.

11 Discrepancy estimate of normal numbers 109 The system (44 is equivalent to the following system of congruences: (52 p j,ν e ν ((n 2 + 1q l 1 l 1 <ν i c j+l1 i<ν 2 m p j,ν e ν (Bq i mod q, j 1,..., i l 1, with 0 n 2 q i l 1 2. For n 2 [0, q i l 1 2] we have the q i l 1 1 distinct vectors of (e l1 +1((n 2 + 1q l 1,..., e i ((n 2 + 1q l 1. Using Lemma 4 by (52 we obtain for n 2 [0, q i l 1 2] the q i l 1 1 distinct vectors of (c l1 +1,..., c i. Let G (g 1,..., g l1, d 1 ((n 2 + 1q l 1 + Bq i,..., d i l1 ((n 2 + 1q l 1 + Bq i 0 n 2 q i l 1 2}. From (42, (44 (52 we find that #G q i l 1 1, from (46 (48 (51 that #(F G q i 1. Hence from (36 the set α ki (n n [Bq i, (B + 1q i 2]} coincides with q i 1 distinct values of j/q i with j [0, q i. By (14 we get A(f/q i, Bq i, q i, (α ki (n n 0 f + 2ε with ε < 1. Hence from (40 we have the assertion of Lemma 5. Corollary 1. (53 A(γ, Bq i, q i, αq n m+2 m n+k } n 0 γq i + 4ε with ε < 1. P r o o f. Analogously to (16, from (14 (35 we have ( f 1 A, Bq i, q i, (α ki (n n 0 A(γ, Bq i, q i, αq n m+2 m n+k } n 0 q i with f [γq i ]. By using Lemma 5 we obtain (53. (54 (55 Corollary 2. Let 1 N < 2 m q 2m. Then A((f + 1/q i, Bq i, q i, (α ki (n n 0 A(γ, n m, N, αq n } n 0 γn + 5qε2 2m with ε < 1, A(γ, n m, 2 m q 2m, αq n } n 0 γ2 m q 2m + 5ε2 m with ε < 1. P r o o f. Let N [N/2 m ], N N 2 m N, N 2 m 1 i0 b i q i with b i, j 1 (56 N 0 0, N j b 2 m iq 2m i, j 1, 2,..., B i N 2 m i 1/q i. i0

12 110 M. B. Levin It is evident that B i (i 1, 2,... are integers, N [0, 2 m. As in (15 we see from (14 that A(γ, n m, N, αq n } n 0 εn 2 + A(γ, N, αq n m+2 m n+k } n 0 Using (56 we have 2 m i1 2 m 1 i0 2 m 1 i0 2 m k1 A(γ, N, αq n m+2 m n+k } n 0, A(γ, N i 1, b 2 m iq 2m i, αq n m+2 m n+k } n 0 A(γ, N 2 m i 1, b i q i, αq n m+2 m n+k } n 0 b i 1 B0 A(γ, n m, N, αq n } n 0 ε2 m + 2 m k1 Applying (53 we obtain 2 m 1 i0 A(γ, N 2 m i 1 + Bq i, q i, αq n m+2 m n+k } n 0. b i 1 B0 A(γ, n m, N, αq n } n 0 ε2 m + A(γ, (B i + Bq i, q i, αq n m+2 m n+k } n 0. 2 m k1 2 m 1 i0 b i 1 (γq i + 4ε i γn + 5qε 1 2 2m B0 with ε 1 1. Assertion (54 is proved. We prove (55 analogously. End of the proof of Theorem 2. For every N q there exists an integer k such that N [n k, n k+1. By (9, this yields N n k +R with 0 R < 2 k q 2k, N 2 (k 1 q 2k 1, 2 k 2 log q N. Applying (9, (13, (14, (54 (55 we obtain A(γ, N, αq n } n 0 k 1 m1 A(γ, n m, 2 m q 2m, αq n } n 0 + A(γ, n k, R, αq n } n 0 k 1 (γ2 m q 2m + O(2 m + γr + O(2 2k m1 Thus, by (1, the theorem is proved. γn + O(2 2k γn + O(log 2 N.

13 Discrepancy estimate of normal numbers 111 Acknowledgments. I am very grateful to Professor Meir Smorodinsky for his hospitality, to the referee for his valuable suggestions. References [B] R. Bejian, Sur certaines suites présentant une faible discrépance à l origine, C. R. Acad. Sci. Paris Sér. A 286 (1978, [BH] M. Bicknell V. E. Hoggart, Jr., Unit determinants in generalized Pascal triangles, Fibonacci Quart. 11 (1978, [F] H. Faure, Discrépance de suites associées à un système de numération (en dimension s, Acta Arith. 41 (1982, [G] I. S. Gal L. Gal, The discrepancy of the sequence (2 n x, Indag. Math. 26 (1964, [K1] N. M. Korobov, Numbers with bounded quotient their applications to questions of Diophantine approximation, Izv. Akad. Nauk SSSR Ser. Mat. 19 (1955, [K2], Distribution of fractional parts of exponential function, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 21 (1966, no. 4, [L1] M. B. L e v i n, The distribution of fractional parts of the exponential function, Soviet Math. (Izv. VUZ 21 (1977, no. 11, [L2], On the upper bounds of discrepancy of completely uniform distributed normal sequences, AMS-IMU joint meeting, Jerusalem, Israel, May 24 26, 1995, Abstracts Amer. Math. Soc. 16 (1995, [N] H. Niederreiter, Rom Number Generation Quasi-Monte Carlo Methods, CBMS-NSF Regional Conf. Ser. in Appl. Math. 63, Philadelphia, [P] V. N. Popov, Asymptotic formula for the sum of sums of the elements of the continued fractions for numbers of the form a/p, J. Soviet Math. 17 (1981, [Po] A. G. Postnikov, Arithmetic modeling of rom processes, Proc. Steklov Inst. Math. 57 (1960. [S] J. Schiffer, Discrepancy of normal numbers, Acta Arith. 47 (1986, [So] I. M. S o b o l, Multidimensional Quadrature Formulas Haar Functions, Nauka, Moscow, 1969 (in Russian. School of Mathematical Sciences Tel-Aviv University Tel-Aviv, Israel mlevin@math.tau.ac.il Received on in revised form on (3105